Abstract

Accurate and reliable peak extraction of axial response signals plays a critical role in confocal microscopy. For axial response signal processing, nonlinear fitting algorithms, such as parabolic, Gaussian or sinc2 fitting may cause significant systematic peak extraction errors. Also, existing error compensation methods require a priori knowledge of the full-width-at-half-maximum of the axial response signal, which can be difficult to obtain in practice. In this paper, we propose a generalised error compensation method for peak extraction from axial response signals. This full-width-at-half-maximum-independent method is based on a corrected parabolic fitting algorithm. With the corrected parabolic fitting algorithm, the systematic error of a parabolic fitting is characterised using a differential equation, following which, the error is estimated and compensated by solving this equation with a first-order approximation. We demonstrate, by Monte Carlo simulations and experiments with various axial response signals with symmetrical and asymmetrical forms, that the corrected parabolic fitting algorithm has significant improvements over existing algorithms in terms of peak extraction accuracy and precision.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

2017 (3)

M. Rahlves, B. Roth, and E. Reithmeier, “Confocal signal evaluation algorithms for surface metrology: uncertainty and numerical efficiency,” Appl. Opt. 56(21), 5920–5926 (2017).
[Crossref] [PubMed]

C. Liu, Y. Liu, T. Zheng, J. Tan, and J. Liu, “Monte Carlo based analysis of confocal peak extraction uncertainty,” Meas. Sci. Technol. 28(10), 105016 (2017).
[Crossref]

J. Wang, L. Pagani, R. K. Leach, W. H. Zeng, B. Colosimo, and L. P. Zhou, “Study of weighted fusion methods for the measurement of surface geometry,” Precis. Eng. 47, 111–121 (2017).
[Crossref]

2016 (2)

J. Liu, Y. Wang, K. Gu, X. You, M. Zhang, M. Li, W. Wang, and J. Tan, “Measuring profile of large hybrid aspherical diffractive infrared elements using confocal profilometer,” Meas. Sci. Technol. 27(12), 125011 (2016).
[Crossref]

H. Cui, W. Zhao, Y. Wang, Y. Fan, L. Qiu, and K. Zhu, “Improving spatial resolution of confocal Raman microscopy by super-resolution image restoration,” Opt. Express 24(10), 10767–10776 (2016).
[Crossref] [PubMed]

2015 (1)

J. Tan, C. Liu, J. Liu, and H. Wang, “Sinc2 fitting for height extraction in confocal scanning,” Meas. Sci. Technol. 27(2), 025006 (2015).
[Crossref]

2014 (2)

2013 (1)

2012 (2)

J. Yang, L. Qiu, W. Zhao, and H. Wu, “Laser differential reflection-confocal focal-length measurement,” Opt. Express 20(23), 26027–26036 (2012).
[Crossref] [PubMed]

F. Mauch, W. Lyda, M. Gronle, and W. Osten, “Object depending artifacts in confocal measurements,” Proc. SPIE 8466, 846609 (2012).
[Crossref]

2010 (2)

H. Jia, J. Yang, and X. Li, “Minimum variance unbiased subpixel centroid estimation of point image limited by photon shot noise,” J. Opt. Soc. Am. A 27(9), 2038–2045 (2010).
[Crossref] [PubMed]

J. Liu, J. Tan, T. Liu, S. Wang, and Y. Cao, “Axial performance parameters developed for analytical design of center shaded filter in high aperture focusing system,” Opt. Commun. 283(21), 4190–4193 (2010).
[Crossref]

2009 (1)

B. S. Chun, K. Kim, and D. Gweon, “Three-dimensional surface profile measurement using a beam scanning chromatic confocal microscope,” Rev. Sci. Instrum. 80(7), 073706 (2009).
[Crossref] [PubMed]

2004 (1)

2002 (1)

1998 (1)

B. V. R. Tata and B. Raj, “Confocal laser scanning microscopy: Applications in material science and technology,” Bull. Mater. Sci. 21(4), 263–278 (1998).
[Crossref]

1989 (1)

T. Wilson and A. R. Carlini, “The effect of aberrations on the axial response of confocal imaging systems,” J. Microsc. 154(3), 243–256 (1989).
[Crossref]

1987 (1)

1986 (1)

Brill, F.

Cao, Y.

J. Liu, J. Tan, T. Liu, S. Wang, and Y. Cao, “Axial performance parameters developed for analytical design of center shaded filter in high aperture focusing system,” Opt. Commun. 283(21), 4190–4193 (2010).
[Crossref]

Carlini, A. R.

T. Wilson and A. R. Carlini, “The effect of aberrations on the axial response of confocal imaging systems,” J. Microsc. 154(3), 243–256 (1989).
[Crossref]

T. Wilson and A. R. Carlini, “Size of the detector in confocal imaging systems,” Opt. Lett. 12(4), 227–229 (1987).
[Crossref] [PubMed]

Chen, C.

Chou, C. H.

Chun, B. S.

B. S. Chun, K. Kim, and D. Gweon, “Three-dimensional surface profile measurement using a beam scanning chromatic confocal microscope,” Rev. Sci. Instrum. 80(7), 073706 (2009).
[Crossref] [PubMed]

Colosimo, B.

J. Wang, L. Pagani, R. K. Leach, W. H. Zeng, B. Colosimo, and L. P. Zhou, “Study of weighted fusion methods for the measurement of surface geometry,” Precis. Eng. 47, 111–121 (2017).
[Crossref]

Corle, T. R.

Cui, H.

Fan, Y.

Grewe, A.

Gronle, M.

F. Mauch, W. Lyda, M. Gronle, and W. Osten, “Object depending artifacts in confocal measurements,” Proc. SPIE 8466, 846609 (2012).
[Crossref]

Gu, K.

J. Liu, Y. Wang, K. Gu, X. You, M. Zhang, M. Li, W. Wang, and J. Tan, “Measuring profile of large hybrid aspherical diffractive infrared elements using confocal profilometer,” Meas. Sci. Technol. 27(12), 125011 (2016).
[Crossref]

Gweon, D.

B. S. Chun, K. Kim, and D. Gweon, “Three-dimensional surface profile measurement using a beam scanning chromatic confocal microscope,” Rev. Sci. Instrum. 80(7), 073706 (2009).
[Crossref] [PubMed]

Hillenbrand, M.

Jia, H.

Jiang, X. J.

Jiang, X. Q.

J. Wang, L. Pagani, L. P. Zhou, X. J. Liu, W. L. Lu, R. K. Leach, and X. Q. Jiang, “Uncertainty-guided intelligent sampling strategy for high-efficiency surface measurement via free-knot B-spline regression modelling,” Precis. Eng., In press,
[Crossref]

Kim, K.

B. S. Chun, K. Kim, and D. Gweon, “Three-dimensional surface profile measurement using a beam scanning chromatic confocal microscope,” Rev. Sci. Instrum. 80(7), 073706 (2009).
[Crossref] [PubMed]

Kino, G. S.

Leach, R. K.

J. Wang, L. Pagani, R. K. Leach, W. H. Zeng, B. Colosimo, and L. P. Zhou, “Study of weighted fusion methods for the measurement of surface geometry,” Precis. Eng. 47, 111–121 (2017).
[Crossref]

J. Wang, L. Pagani, L. P. Zhou, X. J. Liu, W. L. Lu, R. K. Leach, and X. Q. Jiang, “Uncertainty-guided intelligent sampling strategy for high-efficiency surface measurement via free-knot B-spline regression modelling,” Precis. Eng., In press,
[Crossref]

Li, M.

J. Liu, Y. Wang, K. Gu, X. You, M. Zhang, M. Li, W. Wang, and J. Tan, “Measuring profile of large hybrid aspherical diffractive infrared elements using confocal profilometer,” Meas. Sci. Technol. 27(12), 125011 (2016).
[Crossref]

Li, X.

Li, Z.

Liu, C.

C. Liu, Y. Liu, T. Zheng, J. Tan, and J. Liu, “Monte Carlo based analysis of confocal peak extraction uncertainty,” Meas. Sci. Technol. 28(10), 105016 (2017).
[Crossref]

J. Tan, C. Liu, J. Liu, and H. Wang, “Sinc2 fitting for height extraction in confocal scanning,” Meas. Sci. Technol. 27(2), 025006 (2015).
[Crossref]

Liu, D.

Liu, J.

C. Liu, Y. Liu, T. Zheng, J. Tan, and J. Liu, “Monte Carlo based analysis of confocal peak extraction uncertainty,” Meas. Sci. Technol. 28(10), 105016 (2017).
[Crossref]

J. Liu, Y. Wang, K. Gu, X. You, M. Zhang, M. Li, W. Wang, and J. Tan, “Measuring profile of large hybrid aspherical diffractive infrared elements using confocal profilometer,” Meas. Sci. Technol. 27(12), 125011 (2016).
[Crossref]

J. Tan, C. Liu, J. Liu, and H. Wang, “Sinc2 fitting for height extraction in confocal scanning,” Meas. Sci. Technol. 27(2), 025006 (2015).
[Crossref]

J. Liu, J. Tan, T. Liu, S. Wang, and Y. Cao, “Axial performance parameters developed for analytical design of center shaded filter in high aperture focusing system,” Opt. Commun. 283(21), 4190–4193 (2010).
[Crossref]

Liu, T.

J. Liu, J. Tan, T. Liu, S. Wang, and Y. Cao, “Axial performance parameters developed for analytical design of center shaded filter in high aperture focusing system,” Opt. Commun. 283(21), 4190–4193 (2010).
[Crossref]

Liu, X.

Liu, X. J.

J. Wang, L. Pagani, L. P. Zhou, X. J. Liu, W. L. Lu, R. K. Leach, and X. Q. Jiang, “Uncertainty-guided intelligent sampling strategy for high-efficiency surface measurement via free-knot B-spline regression modelling,” Precis. Eng., In press,
[Crossref]

Liu, Y.

C. Liu, Y. Liu, T. Zheng, J. Tan, and J. Liu, “Monte Carlo based analysis of confocal peak extraction uncertainty,” Meas. Sci. Technol. 28(10), 105016 (2017).
[Crossref]

Lu, W.

Lu, W. L.

J. Wang, L. Pagani, L. P. Zhou, X. J. Liu, W. L. Lu, R. K. Leach, and X. Q. Jiang, “Uncertainty-guided intelligent sampling strategy for high-efficiency surface measurement via free-knot B-spline regression modelling,” Precis. Eng., In press,
[Crossref]

Lyda, W.

F. Mauch, W. Lyda, M. Gronle, and W. Osten, “Object depending artifacts in confocal measurements,” Proc. SPIE 8466, 846609 (2012).
[Crossref]

Mauch, F.

F. Mauch, W. Lyda, M. Gronle, and W. Osten, “Object depending artifacts in confocal measurements,” Proc. SPIE 8466, 846609 (2012).
[Crossref]

Mitschunas, B.

Osten, W.

F. Mauch, W. Lyda, M. Gronle, and W. Osten, “Object depending artifacts in confocal measurements,” Proc. SPIE 8466, 846609 (2012).
[Crossref]

Pagani, L.

J. Wang, L. Pagani, R. K. Leach, W. H. Zeng, B. Colosimo, and L. P. Zhou, “Study of weighted fusion methods for the measurement of surface geometry,” Precis. Eng. 47, 111–121 (2017).
[Crossref]

J. Wang, L. Pagani, L. P. Zhou, X. J. Liu, W. L. Lu, R. K. Leach, and X. Q. Jiang, “Uncertainty-guided intelligent sampling strategy for high-efficiency surface measurement via free-knot B-spline regression modelling,” Precis. Eng., In press,
[Crossref]

Qiu, L.

Rahlves, M.

Raj, B.

B. V. R. Tata and B. Raj, “Confocal laser scanning microscopy: Applications in material science and technology,” Bull. Mater. Sci. 21(4), 263–278 (1998).
[Crossref]

Reithmeier, E.

Roth, B.

Ruprecht, A. K.

Shao, R.

Sheng, Z.

Sinzinger, S.

Tan, J.

C. Liu, Y. Liu, T. Zheng, J. Tan, and J. Liu, “Monte Carlo based analysis of confocal peak extraction uncertainty,” Meas. Sci. Technol. 28(10), 105016 (2017).
[Crossref]

J. Liu, Y. Wang, K. Gu, X. You, M. Zhang, M. Li, W. Wang, and J. Tan, “Measuring profile of large hybrid aspherical diffractive infrared elements using confocal profilometer,” Meas. Sci. Technol. 27(12), 125011 (2016).
[Crossref]

J. Tan, C. Liu, J. Liu, and H. Wang, “Sinc2 fitting for height extraction in confocal scanning,” Meas. Sci. Technol. 27(2), 025006 (2015).
[Crossref]

J. Liu, J. Tan, T. Liu, S. Wang, and Y. Cao, “Axial performance parameters developed for analytical design of center shaded filter in high aperture focusing system,” Opt. Commun. 283(21), 4190–4193 (2010).
[Crossref]

Tata, B. V. R.

B. V. R. Tata and B. Raj, “Confocal laser scanning microscopy: Applications in material science and technology,” Bull. Mater. Sci. 21(4), 263–278 (1998).
[Crossref]

Tiziani, H. J.

Wang, H.

J. Tan, C. Liu, J. Liu, and H. Wang, “Sinc2 fitting for height extraction in confocal scanning,” Meas. Sci. Technol. 27(2), 025006 (2015).
[Crossref]

Wang, J.

C. Chen, J. Wang, X. Liu, W. Lu, H. Zhu, and X. J. Jiang, “Influence of sample surface height for evaluation of peak extraction algorithms in confocal microscopy,” Appl. Opt. 57(22), 6516–6526 (2018).
[Crossref] [PubMed]

J. Wang, L. Pagani, R. K. Leach, W. H. Zeng, B. Colosimo, and L. P. Zhou, “Study of weighted fusion methods for the measurement of surface geometry,” Precis. Eng. 47, 111–121 (2017).
[Crossref]

J. Wang, L. Pagani, L. P. Zhou, X. J. Liu, W. L. Lu, R. K. Leach, and X. Q. Jiang, “Uncertainty-guided intelligent sampling strategy for high-efficiency surface measurement via free-knot B-spline regression modelling,” Precis. Eng., In press,
[Crossref]

Wang, S.

J. Liu, J. Tan, T. Liu, S. Wang, and Y. Cao, “Axial performance parameters developed for analytical design of center shaded filter in high aperture focusing system,” Opt. Commun. 283(21), 4190–4193 (2010).
[Crossref]

Wang, W.

J. Liu, Y. Wang, K. Gu, X. You, M. Zhang, M. Li, W. Wang, and J. Tan, “Measuring profile of large hybrid aspherical diffractive infrared elements using confocal profilometer,” Meas. Sci. Technol. 27(12), 125011 (2016).
[Crossref]

Wang, Y.

J. Liu, Y. Wang, K. Gu, X. You, M. Zhang, M. Li, W. Wang, and J. Tan, “Measuring profile of large hybrid aspherical diffractive infrared elements using confocal profilometer,” Meas. Sci. Technol. 27(12), 125011 (2016).
[Crossref]

H. Cui, W. Zhao, Y. Wang, Y. Fan, L. Qiu, and K. Zhu, “Improving spatial resolution of confocal Raman microscopy by super-resolution image restoration,” Opt. Express 24(10), 10767–10776 (2016).
[Crossref] [PubMed]

Wiesendanger, T. F.

Wilson, T.

T. Wilson and A. R. Carlini, “The effect of aberrations on the axial response of confocal imaging systems,” J. Microsc. 154(3), 243–256 (1989).
[Crossref]

T. Wilson and A. R. Carlini, “Size of the detector in confocal imaging systems,” Opt. Lett. 12(4), 227–229 (1987).
[Crossref] [PubMed]

Wu, H.

Yang, J.

You, X.

J. Liu, Y. Wang, K. Gu, X. You, M. Zhang, M. Li, W. Wang, and J. Tan, “Measuring profile of large hybrid aspherical diffractive infrared elements using confocal profilometer,” Meas. Sci. Technol. 27(12), 125011 (2016).
[Crossref]

Zeng, W. H.

J. Wang, L. Pagani, R. K. Leach, W. H. Zeng, B. Colosimo, and L. P. Zhou, “Study of weighted fusion methods for the measurement of surface geometry,” Precis. Eng. 47, 111–121 (2017).
[Crossref]

Zhang, M.

J. Liu, Y. Wang, K. Gu, X. You, M. Zhang, M. Li, W. Wang, and J. Tan, “Measuring profile of large hybrid aspherical diffractive infrared elements using confocal profilometer,” Meas. Sci. Technol. 27(12), 125011 (2016).
[Crossref]

Zhao, W.

Zheng, T.

C. Liu, Y. Liu, T. Zheng, J. Tan, and J. Liu, “Monte Carlo based analysis of confocal peak extraction uncertainty,” Meas. Sci. Technol. 28(10), 105016 (2017).
[Crossref]

Zhou, L. P.

J. Wang, L. Pagani, R. K. Leach, W. H. Zeng, B. Colosimo, and L. P. Zhou, “Study of weighted fusion methods for the measurement of surface geometry,” Precis. Eng. 47, 111–121 (2017).
[Crossref]

J. Wang, L. Pagani, L. P. Zhou, X. J. Liu, W. L. Lu, R. K. Leach, and X. Q. Jiang, “Uncertainty-guided intelligent sampling strategy for high-efficiency surface measurement via free-knot B-spline regression modelling,” Precis. Eng., In press,
[Crossref]

Zhu, H.

Zhu, K.

Appl. Opt. (5)

Bull. Mater. Sci. (1)

B. V. R. Tata and B. Raj, “Confocal laser scanning microscopy: Applications in material science and technology,” Bull. Mater. Sci. 21(4), 263–278 (1998).
[Crossref]

J. Microsc. (1)

T. Wilson and A. R. Carlini, “The effect of aberrations on the axial response of confocal imaging systems,” J. Microsc. 154(3), 243–256 (1989).
[Crossref]

J. Opt. Soc. Am. A (1)

Meas. Sci. Technol. (3)

J. Liu, Y. Wang, K. Gu, X. You, M. Zhang, M. Li, W. Wang, and J. Tan, “Measuring profile of large hybrid aspherical diffractive infrared elements using confocal profilometer,” Meas. Sci. Technol. 27(12), 125011 (2016).
[Crossref]

J. Tan, C. Liu, J. Liu, and H. Wang, “Sinc2 fitting for height extraction in confocal scanning,” Meas. Sci. Technol. 27(2), 025006 (2015).
[Crossref]

C. Liu, Y. Liu, T. Zheng, J. Tan, and J. Liu, “Monte Carlo based analysis of confocal peak extraction uncertainty,” Meas. Sci. Technol. 28(10), 105016 (2017).
[Crossref]

Opt. Commun. (1)

J. Liu, J. Tan, T. Liu, S. Wang, and Y. Cao, “Axial performance parameters developed for analytical design of center shaded filter in high aperture focusing system,” Opt. Commun. 283(21), 4190–4193 (2010).
[Crossref]

Opt. Express (3)

Opt. Lett. (3)

Precis. Eng. (1)

J. Wang, L. Pagani, R. K. Leach, W. H. Zeng, B. Colosimo, and L. P. Zhou, “Study of weighted fusion methods for the measurement of surface geometry,” Precis. Eng. 47, 111–121 (2017).
[Crossref]

Proc. SPIE (1)

F. Mauch, W. Lyda, M. Gronle, and W. Osten, “Object depending artifacts in confocal measurements,” Proc. SPIE 8466, 846609 (2012).
[Crossref]

Rev. Sci. Instrum. (1)

B. S. Chun, K. Kim, and D. Gweon, “Three-dimensional surface profile measurement using a beam scanning chromatic confocal microscope,” Rev. Sci. Instrum. 80(7), 073706 (2009).
[Crossref] [PubMed]

Other (7)

T. Wilson, Confocal Microscopy (Academic, 1990).

R. Leach, Optical Measurement of Surface Topography (Springer, 2011).

M. Gu, Principles of three dimensional imaging in confocal microscopes (World Scientific, 1996).

J. Wang, L. Pagani, L. P. Zhou, X. J. Liu, W. L. Lu, R. K. Leach, and X. Q. Jiang, “Uncertainty-guided intelligent sampling strategy for high-efficiency surface measurement via free-knot B-spline regression modelling,” Precis. Eng., In press,
[Crossref]

H. H. Hopkins, Wave Theory of Aberrations (Oxford, Clarendon Press, 1950).

J. Liu and J. Tan, Confocal Microscopy (Morgan & Claypool, 2016).

BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML. Evaluation of Measurement Data—Supplement 1 to the ‘Guide to the Expression of Uncertainty in Measurement’—Propagation of Distributions using a Monte Carlo Method. (Joint Committee for Guides in Metrology, Bureau International des Poids et Mesures, 2008).

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Figures (13)

Fig. 1
Fig. 1 Graph of the thresholding technique for the sampled ARSs at certain sample surface heights. The abscissa is normalised to ΔU.
Fig. 2
Fig. 2 Height-dependent systematic errors at different sample surface heights for PFA ( ΔU= 1 9 FWHM,T=0.45).
Fig. 3
Fig. 3 C 1 and C 2 versus sample surface height X( ΔU= 1 9 FWHM,T=0.45).
Fig. 4
Fig. 4 Height-dependent systematic errors at different sample surface heights for PFA, GFA and CPFA ( ΔU= 1 9 FWHM,T=0.45).
Fig. 5
Fig. 5 Performance of CPFA at different surface heights ( ΔU= 1 9 FWHM,T=0.45). (a) Systematic peak extraction errors at different sample surface heights and (b) Peak extraction standard deviations at different sample surface heights. Here the mean of peak extraction errors in Fig. 5 has the same meaning as the systematic error.
Fig. 6
Fig. 6 Performance of CPFA with different scanning steps (high noise level). (a) RMS of systematic errors with different scanning steps and (b) RMS of standard deviations with different scanning steps.
Fig. 7
Fig. 7 Performance of CPFA with different scanning steps (medium noise level). (a) RMS of systematic errors with different scanning steps and (b) RMS of standard deviations with different scanning steps
Fig. 8
Fig. 8 Performance of CPFA with different scanning steps (low noise level). (a) RMS of systematic errors with different scanning steps and (b) RMS of standard deviations with different scanning steps
Fig. 9
Fig. 9 Asymmetrical ARS with combined aberrations ( W 040 =0.3, W 131 =0, W 222 =0.4).
Fig. 10
Fig. 10 Performance of CPFA with different scanning steps under the asymmetrical ARS (low noise level). (a) RMS of systematic peak extraction errors with different scanning steps and (b) RMS of peak extraction standard deviations with different scanning steps.
Fig. 11
Fig. 11 Schematic of the experimental CCM.
Fig. 12
Fig. 12 The measured spectrometer signal of the CCM.
Fig. 13
Fig. 13 Performance comparisons between CPFA and other fitting algorithms including PFA, GFA and SFA. (a) Local height deviations at different surface heights and (b) Height extraction standard deviations at different surface heights.

Tables (1)

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Table 1 Noise settings of simulations

Equations (23)

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U =[ U -n , U -n+1 ,..., U -1 ,0, U 1 ,..., U m-1 , U m ],
I( U j |X )= [ sin[ ( U j X )/2 ] ( U j X )/2 ] 2 ,
Q= [ I( U j |X ) a 0 a 1 U j a 2 U j 2 ] 2 min.
p( X )= a 1 2 a 2 ,
a 1 =[ U j I( U j |X ) ] [ U j 2 ] 2 [ I( U j |X ) ][ U j 2 ][ U j 3 ] +[ I( U j |X ) ][ U j ][ U j 4 ][ U j 2 I( U j |X ) ][ U j ][ U j 2 ], [ U j I( U j |X ) ][ 1 ][ U j 4 ]+[ U j 2 I( U j |X ) ][ 1 ][ U j 3 ]
a 2 =[ U j 2 I( U j |X ) ] [ U j ] 2 [ U j I( U j |X ) ][ U j ][ U j 2 ] [ I( U j |X ) ][ U j ][ U j 3 ]+[ I( U j |X ) ] [ U j 2 ] 2 . [ U j 2 I( U j |X ) ][ 1 ][ U j 2 ]+[ U j I( U j |X ) ][ 1 ][ U j 3 ]
e X = C 1 C 2 X C 2 e1,
C 1 = ( a 1 / X ) 2 a 2 ,
C 2 = ( a 2 / X ) a 2 .
e( X )( C 1 1 )X
e( 0 )=0.
D I = [ I( U j |X )I( U j |X ) ] ,
I( U j |X )I( U j |X )X[ I'( U j |0 )I'( U j |0 ) ],
I'( U j |0 ) 1 2 [ I'( U j |X )+I'( U j | X ) ].
e c ^ ( X )( C 1 1 )[ D I [ I'( U j |0 )I'( U j |0 ) ] ].
p c ( X )=p( X ) e c ^ ( X ).
p r ( X )=A[ U , I ( U + N U |X )+ N I ],
e r ( X )= p r ( X )X.
RMS{ E[ e r ( X ) ] }= 1 ΔU 0.5ΔU 0.5ΔU { E[ e r ( X ) ] } 2 dX ,
RMS{ STD[ e r ( X ) ] }= 1 ΔU 0.5ΔU 0.5ΔU { STD[ e r ( X ) ] } 2 dX .
I( U j |X )= | 2 2π 0 1 0 2π P j ( ρ,θ ) P j ( ρ,πθ )dθρdρ | 2
P j ( ρ,θ )={ exp[ 1 ( U j X ) 2 ρ 2 ]exp[ 1 W( ρ,θ ) ] ρ1 0 ρ>1
W( ρ,θ )=2π( W 040 ρ 4 + W 131 ρ 3 cosθ+ W 222 ρ 2 cos 2 θ )

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